Direct dark matter searches - Test of the Big Bounce Cosmology

We consider the possibility of using dark matter particle's mass and its interaction cross section as a smoking gun signal of the existence of a Big Bounce at the early stage in the evolution of our currently observed universe. A study of dark matter production in the pre-bounce contraction and the post bounce expansion epochs of this universe reveals a new venue for achieving the observed relic abundance of our present universe. Specifically, it predicts a characteristic relation governing a dark matter mass and interaction cross section and a factor of $1/2$ in thermally averaged cross section, as compared to the non-thermal production in standard cosmology, is needed for creating enough dark matter particle to satisfy the currently observed relic abundance because dark matter is being created during the pre-bounce contraction, in addition to the post-bounce expansion. As the production rate is lower than the Hubble expansion rate information of the bounce universe evolution is preserved. Therefore once the value of dark matter mass and interaction cross section are obtained by direct detection in laboratories, this alternative route becomes a signature prediction of the bounce universe scenario. This leads us to consider a scalar dark matter candidate, which if it is light, has important implications on dark matter searches.

can be directly detected mainly via the recoiling of a nucleus (A,Z) in elastic scattering. The event rate for such a process can be computed from the following ingredients [14]: i) The elementary nucleon cross section. ii) knowledge of the relevant nuclear matrix elements obtained with as reliable as possible many body nuclear wave functions, iii) knowledge of the WIMP density in our vicinity and its velocity distribution.
In the standard nuclear recoil experiments, first proposed more than 30 years ago [15], one has to face the problem that the reaction of interest does not have a characteristic feature to distinguish it from the background. So for the expected low counting rates the background is a formidable problem. Some special features of the WIMP-nuclear interaction can be exploited to reduce the background problems. Such are: i) the modulation effect: this yields a periodic signal due to the motion of the earth around the sun. Unfortunately this effect, also proposed a long time ago [16] and subsequently studied by many authors [14,[17][18][19][20][21][22][23][24][25], is small and becomes even smaller than 2% due to cancelations arising from nuclear physics effects, ii) backward-forward asymmetry expected in directional experiments, i.e. experiments in which the direction of the recoiling nucleus is also observed. Such an asymmetry has also been predicted a long time ago [26], but it has not been exploited, since such experiments have been considered very difficult to perform. Some progress has, however, has recently been made in this direction and they now appear feasible [26][27][28][29][30][31][32][33][34][35][36][37][38]. In such experiments the event rate depends on the direction of observation. In the most favorable direction, opposite to the sun's direction of motion, is comparable to the standard event rate. The sensitivity of these experiments for various halo models has also been discussed [31,32]. Furthermore we should mention that in such experiments [27,30,38] all events are counted. If some interesting events can be found, they can be established by further analyzing them by the direction of the observed recoils.
An essential ingredient in direct WIMP detection is the WIMP density in our vicinity and, especially, the WIMP velocity distribution. The dark matter in the solar neighborhood is commonly assumed to be smoothly distributed in space and to have a Maxwellian velocity distribution. Some of the calculations have considered various forms of phenomenological non symmetric velocity distributions [39][40][41][42] [23,30,31] and some of them even more exotic dark matter flows like the late infall of dark matter into the galaxy, i.e caustic rings [43][44][45][46][47] and Sagittarius dark matter [48].
In addition to the above models very recently it was found that the velocity distributions measured in high resolution numerical simulations exhibit deviations from the standard Maxwell-Boltzmann assumption, especially at large velocities [49,50]. Furthermore a distinction was between a velocity structure that is spatially localized, such as streams [51,52], and that which is spatially homogenized, which was designated as "debris flow" [53]. Both streams and debris flows arise from the disruption of satellites that fall into the Milky Way, but differ in the relative amount of phase-mixing that they have undergone. Implications of streams [54] and the debris flows in direct dark matter searches have also been considered [55], [56].
In the present paper we will address the following points: • The implications scalar WIMPs on the expected event rates. The interest in such a WIMP has recently been revived due to a new scenario of dark matter production in bounce cosmology [57,58] in which the authors point out the possibility of using dark matter as a probe of a big bounce at the early stage of cosmic evolution. A model independent study of dark matter production in the contraction and expansion phases of the Big Bounce reveals a new venue for achieving the observed relic abundance in which dark matter was produced completely out of chemical equilibrium 1 . A characteristic relation, Fig. 2, comes out of the model independent analysis. This is to be contrasted ≺ συ The cross section ≺ συ as a function of the WIMP mass. In the standard cosmology it is a constant (solid line), but it varies considerably in the bounce universe scenario (dotted line) with the straight line (cross-section being independent of dark matter mass) of the standard cosmology. Once DM mass and its coupling constant with ordinary matter are extracted from experimental data we can check if they obey the predicted relation. In this way, this alternative route of dark matter production in bounce cosmology can be used to test the bounce cosmos hypothesis.
• In order to settle the issues raised above we will compute the differential and total event rates in a variety of targets such as those employed in XENON [60,61], CoGENT [62], DAMA [63,64], LUX [65], CDMSII [66], CRESST [67] and PICASSO [68,69]. For this study we will consider not only the standard Maxwell Boltzmann distribution but also the effects of debris flows [55] on these rates including the modulation due to annual motion of the Earth [70].

Figure 2:
A schematic plot of the time evolution of dark matter in a generic bounce universe scenario. Two pathways of producing dark matter yet satisfying current observations thermal production (which is indistinguishable from standard cosmology) and non-thermal production (characteristic to bounce universe) are illustrated. The horizontal axis indicates both the time, t, as well as the temperature, T, of the cosmological background.
In any case, regardless of the validity of the big bounce universe scenario, the scalar WIMPs have the characteristic feature that the elementary cross section in their scattering off ordinary quarks is increasing as the WIMPs get lighter, which leads to an interesting experimental feature, namely it is expected to enhance the event rates at low WIMP mass. In the present calculation we will adopt this view and study its implications in direct direct dark matter searches compared to other types of WIMPs, such as the neutralinos, which we will call standard. Scalar WIMP's can occur in particle models. Examples are i) In Kaluza-Klein theories for models involving universal extra dimensions (for applications to direct dark matter detection see, e.g., [71]). In such models the scalar WIMPs are characterized by ordinary couplings, but they are expected to be quite massive. ii) very light particles [72] not relevant to the ongoing WIMP searches ii) Scalar WIMPS such as those considered previously in various extensions of the standard model [73], which can be quite light and long lived protected by a discrete symmetry. Thus they are viable cold dark matter candidates.

The big bounce universe scenario
Recently a stable as well as scale-invariant power spectrum of primordial density perturbations is finally obtained [74,75] in the bounce universe scenario. The "Bounce Cosmology" postulates that there exists a phase of matter-dominated contraction before the Big Bang [76] during which the matter content of the universe comes into thermal contact-resulting in a scale invariant spectrum-before a subsequent expansion after the big bounce. In view of this development we are motivated to work out further experimental or observational predictions the Bounce Universe model [57,58] (See also [77].) 2 . Our study is model independent of a particular bounce model and our predictions are of particle physics nature and can be tested independently at LHC or dark matter direct detections, outside of the cosmological context. A signature prediction from the bounce universe: By investigating the production process of dark matter in the pre-bounce contraction and the post-bounce expansion epochs of a generic bounce universe, we find that, in the big bounce scenario, dark matter production can be extended beyond the Big Bang, as shown in Fig. 1 (compare the dotted and solid lines). Furthermore an out-of-thermal-equilibrium production of dark matter is allowed, which encodes information of early universe evolution, marked the non-thermal production" in Fig. 2. Specifically it predicts a relation governing a dark matter mass and interaction cross section, depicted by the solid line in Fig. 1 . This behavior reflects a mass dependence of the cross-section, characteristic of a scalar type WIMP. As shown in Fig. 2, we divide the bounce (See [92,93] for recent reviews.) schematically into three stages to facilitate a model independent analysis [57,58]. 3 The particle model.
If the WIMP is a scalar [94][95][96][97] particle χ interacting with another scalar φ via a quartic coupling the cross section ≺ συ for the process: in the center of mass system is given by: In the limit in which m φ >> m χ and √ s ≈ 2m φ we find: We will assume in this work that φ is the Higgs scalar discovered at LHC.
If the WIMP is a scalar particle χ interacting with another scalar φ via a quartic coupling the cross section ≺ συ for the process: in the center of mass system is given by: In the limit in which m φ >> m χ and √ s ≈ 2m φ we find: which is in essential agreement with the expression after Eq. (4) given previously [57,58], but with different assumptions. For the scalar WIMP-quark scattering the relevant Feynman diagram is shown in Fig. 3.  The resulting nucleon cross section is given by: Note that the vacuum expectation value ≺ φ 0 in the quartic coupling is canceled by the Yukawa coupling of the Higgs with the quarks. The parameter f q is related to the probability of finding the quark q in the nucleon: i.e. the heavy quarks become important, even though the probability of finding them in the nucleon is small. If the scalar is the Higgs particle discovered at LHC, λ = 1/2, m φ = 126 GeV, one finds: The value of q f q , of course, can vary, but a reasonable, albeit rather optimistic, value of 0.5 is acceptable [98,99], [100]. Thus This for m χ = 50GeV this value is quite a bit bigger than the limit extracted from the current experimental searches. So in our treatment we have fixed the parameter σ 0 in the nucleon cross section so that for a WIMP mass of 50 GeV we get the limit extracted from experiments, e.g. 10 −8 pb from XENON100 [61,101]. The thus obtained cross section is exhibited in Fig. 4. It is interesting to compare the behavior of this cross section with that of the relic abundance of the BUS shown in Fig. 1. We note that this mass dependence of the cross section of scalar WIMPs, i.e. exhibiting an enhancement in the low WIMP mass regime, may favor the searches at low energy transfers.
In the case of light WIMPs, another interesting domain of the BUS (Fig. 1), one finds that WIMPs with energy less than 100 MeV cannot produce a detectable recoiling nucleus, but they could produce electrons [102] with energies in the tens of eV, which could be detected with current mixed phase detectors [103]. We are not, however, going to discuss further this possibility in this work. If the WIMP is a scalar particle, however, it can interact in a similar pattern with other fermions, e.g. electrons. The relevant Feynman diagram is shown in Fig.  3.
For WIMPs with mass in the range of the electron mass, both the WIMP and the electron are not relativistic. So the expression for elementary electron cross section is similar to that of hadrons , i.e. it is now given by: (3.11) obtained using the same values of λ and φ as above. This is a respectable size cross section dependent on the ratio m χ /m e . In this case one must consider electron recoils, but the highest possible electron energy is about 1.5 eV and the WIMP mass must greater than 0.3 electron masses. So the detection of WIMPs with mass around the electron mass requires another type of detector and it will not be discussed further in the present work.  4 The formalism for the WIMP-nucleus differential event rate The most interesting quantity which depends on the velocity distribution is the quantity g(υ min ). For the M-B distribution in the local frame it is defined as follows: For isotropic debris flows [55] it is given by: In what follows we will find it useful to expand g(υ min , υ E (α)) in powers of δ, the ratio of the Earth's velocity around the sun divided by the velocity υ 0 of the sun around the galaxy (220km/s). Keeping terms up to linear in δ ≈ 0.135 and expressing everything in dimensionless variables we obtain: where Ψ 0 (x represents the quantity relevant for the average rate , Ψ 1 (x, which is proportional to δ, represents the modulation and α is the phase of the Earth (α = 0 around June 3nd). In the case of the flows they were derived from the semi-analytic approximations of simulations as discussed by Spergel and co-workers [55].
In the case of a M-B distribution these functions have been given previously [104]. For isotropic debris flows one finds: We note that the variable x depends on the nuclear recoil energy E R as well as the WIMPnucleus reduced mass. As we shall see below there is an additional dependence of the rates on E R coming from the nuclear form factor.
At Earth-frame velocities greater than 450 km/s, debris flow comprises more than half of the dark matter at the Sun's location, and up to 80% at even higher velocities [55]. In the VL2 simulation, the combination of debris flows and standard M-B is very well fit by the function This function is exhibited in Fig. 6. In this case we find: The functions Ψ 0 (x) and Ψ 1 (x) are exhibited in Fig. 7. As expected in the case of the flows Ψ 0 (x) falls off linearly for large values of x. Note that in all cases Ψ 1 (x) takes both positive and negative values, which affects the location of the maximum of the modulated rate as a function of α, depending on the target and the WIMP mass. We will explore this effect of the different distributions in direct experiments searching any time dependence of the rates. Once these functions are known the formalism to obtain the direct detection rates is fairly well known (see e.g. the recent reviews [105,106]). So we will briefly discuss its essential elements here. The differential event rate can be cast in the form: where the first term represents the time averaged (non modulated) differential event rate, while the second gives the time dependent (modulated) one due to the motion of the Earth (see below). Furthermore with µ r (µ p ) the WIMP-nucleus (nucleon) reduced mass, A is the nuclear mass number and σ n is the elementary WIMP-nucleon cross section. m χ is the WIMP mass and m t the mass of the target. Furthermore one can show that with a = ( √ 2µ r bυ 0 ) −1 , υ 0 the velocity of the sun around the center of the galaxy and b the nuclear harmonic oscillator size parameter characterizing the nuclear wave function. u is the energy transfer Q in dimensionless units given by and F (u) is the nuclear form factor. In the present calculation they were obtained in context of the nuclear shell model in the spirit of [107] (for the spin induced process see,e.g. [107,108]). The form factor is important in the case of a heavy target and large WIMP mass, i.e. for large recoil energies (see Fig. 8).
Note that the parameter a depends both on the WIMP , the target and the velocity distribution. Note also that for a given energy transfer E R the quantity u depends on A. Sometimes one writes the differential rate as: In this formulation H(a E R /Q 0 (A)), the ratio of the modulated to the non modulated differential rate, gives the relative differential modulation amplitude. It coincides with the ratio Ψ 1 (a E R /Q 0 (A))/Ψ 0 (a E R /Q 0 (A)), i.e. it is independent of the nuclear form factor and depends only on the reduced mass and the velocity distribution. It is thus the same for both the coherent and the spin mode. Note that it can take both positive and negative values, which affects the location of the maximum of the modulated rate as a function of α.
For the convenience of the analysis of experiments, however, we will present our results in the form of Eq. 4.9.
5 Some results on differential rates We will apply the above formalism in the case of I and Na, which are components of the target NAI used in the DAMA experiment [63,64] and Ge employed, e.g, by the CoGeNT experiment [62]. The results for the Xe target [60] are similar to those for I and for the 19 F target [68,69] are similar to those for Na . The differential rates dR dQ | A and dH dQ | A , for each component (A = 127 and A = 23) and for A = 73 are exhibited in Fig. 9-20. The nuclear form factor has been included (for a heavy target, like 127 I or 131 Xe, its effect is sizable even for an energy transfer [70] of 10 keV, see Fig. 8).
By comparing the plots of the differential event rates of scalar WIMPs to the standard ones we find that the shapes are the same, but for low mass the scalar WIMPs lead to much larger event rates. So we will restrict the discussion on the shape of these plots to the results obtained for standard WIMPs.
The introduction of debris flows makes a small contribution at low energy transfers. As expected [55] it tends to increase the differential rate at high energy transfers. This is particularly true for light small WIMP-nucleus reduced mass (see Figs 9, 13 and 17). One, however, does not see any particular signature in the shape of the resulting curve. Furthermore the event rate in this region is about five times smaller than the maximum. One, however, observes an interesting pattern concerning the time varying (modulated) part of the rate (see Figs 11,15 and 19). For a heavy target, like 127 I or 131 Xe, it is not surprising that, for WIMPs with relatively large mass, the modulation becomes negative, i.e. the rate becomes minimum in June 3nd, for all models considered here. For low WIMP masses, however, the sign of the modulation due to the flows is opposite to that of the M-B distribution. Thus the use of the light target nucleus 19 F, combined with the low detection threshold of 1.7 keV for recoil nuclei, makes PICASSO particularly sensitive to low mass dark matter particles and gives it also some leverage in the low mass region of the spin independent sector. The present stage of the experiment [109] is approaching the sensitivity to challenge or confirm the claims of seasonal modulations by the DAMA [64] and CoGeNT [62]  , while the result of debris flows is to cause a change in sign as one moves to high energy transfers. Also in this case the modulation amplitude tends to increase as the energy transfer increase, while the corresponding contribution due to the M-B distribution tends to decrease. We should remark though that the total rate (average+modulated) tends to decrease at high momentum transfers. We should also stress that we have presented here the absolute modulate rate (events per kg target per year). The relative modulated amplitude (the ratio of the time varying rate divided by the time averaged) maybe larger.
The above results, as we will see in the next section, have important implications in the total event rates.
Sometimes, as is the case for the DAMA experiment, the target has many components. In such cases the above formalism can be applied as follows:

Some results on total rates
For completeness and comparison we will briefly present our results on the total rates. Integrating the differential rates discussed in the previous section we obtain the total rate R, adding the corresponding time averaged rate R 0 and the total modulated rateH, given by: GeV. Again the asymptotic value at 500 GeV is about 1/5 of the maximum. The situation is very different for a scalar WIMP. At small WIMP masses the event rate becomes huge. The effect will appear less dramatic, if the value of 10 −8 pb is fitted to a much smaller WIMP mass, since it will manifest itself for masses below that choice, but it is there. At high WIMP masses the event rate falls more rapidly with the mass. The relative modulation amplitude, however, being the ratio of the time dependent rate divided by the time averaged rate is the same for both types of WIMPs.
To understand this behavior we should mention that the WIMP mass dependence comes from three sources.
• From the momentum transfer, yielding a contribution to the event rate proportional to the square of µ r (the WIMP-nucleus reduced mass), which vanishes quadratically for zero WIMP mass.
• From the WIMP particle density in our vicinity, which is inversely proportional to the mass (from the rotation curves we infer the density, not the number of particles per unit volume). In the limit of large WIMP mass this wins out over the previous one, since the reduced mass then is essentially the mass of the nucleus. For small WIMP mass the combination of these terms vanishes linearly.
• For scalar WIMPS we have the additional mass dependence coming from the elementary cross section σ n ∝ (1 + m χ /m p ) −2 as we have seen.
We thus conclude that even in the case for a scalar WIMP at a low mass the total rate is proportional to It is clear that, as far as the time average rates R 0 are concerned ,the debris flows do not exhibit any characteristic signature to differentiate them from the standard M-B distribution.
dH/dQ →kg/(y keV) Q →keV Figure 11: The differential rate dH dQ , as a function of the recoil energy for a heavy target, e.g. 127 I assuming a nucleon cross section of 10 −8 pb. Panels (a) (b), (c) and (d) correspond to to 5, 20, 50 and 100 GeV WIMP masses. Otherwise the notation is the same as that of Fig. 5.
The relative modulation amplitude h, however, exhibits a very interesting feature, namely, if caused by the flows, it is negative for all targets, even for the light ones, and in the entire WIMP mass range (minimum in June). On the other hand if it is caused by the M-B distribution it is positive in the case of light targets regardless of the WIMP mass. It is also positive for intermediate/heavy targets, if the WIMPs are relatively light. Then the maximum occurs on June 3nd as expected. It becomes negative only for relatively heavy WIMPs. Thus it is an experimental challenge to measure the small time dependence of the event rate with a relative difference between the maximum and the minimum of 2h ≈ 4%. From such data on both light and heavy targets, if and when they become available, one may may be able: i)to get a hint about the size of the WIMP mass and ii) infer the existence of flows.

Discussion and conclusions
In the present paper we first obtained results on the differential event rates, both modulated and time averaged. We have considered a new type of viable WIMP, namely a scalar WIMP, motivated by the Big Bounce Scenario of Cosmology (BUS). We then compared the obtained results with the standard WIMP with a nucleon cross section independent of the WIMP dH/dQ →kg/(y keV) mass. We found that: • The nucleon cross section is a decreasing function as the WIMP mass increases. This is in line with the predictions of BUS (compare Fig. 1 and Fig. 4).
• The above mass dependence leads to an increase of the rates at a low WIMP mass. This may be good news for the low threshold experiments using light nuclear targets (DM-TPC, NEWAGE, DRIFT,MIMAC etc), which are sensitive to low mass WIMPs.
• The maximum of the total event rate is shifted to a much lower regime, which may require a lower recoil energy threshold than currently achieved.
• As far as we know this behavior of the cross section is not excluded by the current data. In fact it may aid the analysis of the experimental data in the low WIMP mass regime even though there is a tendency for model independent analysis of the data, as e.g. in DAMA/LIBRA [110].
• It may interesting to draw exclusion plots with this new nucleon cross section and extract the value of σ 0 entering Eq. (3.9).
• It may also help explaining the large cross section extracted from the recent CRESST data [67], if they persist.
dR/dQ →kg/(y keV) We also examined the sensitivity of the obtained results to the velocity distribution (the nuclear form factor is the same of both types of WIMPs). We considered both a standard M-B velocity distribution and also models, which extend it, e.g. debris flows, which have also been considered [55], [56] previously. Thus we found out that: • The flows indeed enhance the time averaged rates at relatively high energy transfers compared to the M-B distribution, at the expense of the corresponding rates at low energy transfers . All rates, however, fall as the energy transfer increases. This fall is only partially due to the velocity distribution. It is also caused by the nuclear form factor, in particular in the case of heavy targets. Anyway this behavior cannot be exploited to differentiate between them, since the WIMP mass is not known. Thus the time averaged rates do not provide a clear signature to differentiate the debris flows from the standard M-B distribution.
• The differential time dependent (modulated) rates provide such a signature, the sign of the modulation amplitude, which determines the position of the maximum. At sufficiently low energy transfer both the M-B and the debris flows favor a negative sign (minimum on June 3nd), with the flows insisting on such behavior more strongly and exhibiting it all the way to high energy transfers. So if the flows are there this signature may be seen even with detectors, which do not have a very low energy threshold.
dR/dQ →kg/(y keV) Q →keV Figure 14: The same as in Fig. 10 in the case of a light target, e.g. 23 Na.
• For the M-B distribution this behavior is manifested for an energy which depends on the target and the WIMP mass ( see Figs 11,12,15,16,19,20). Thus e.g. for a heavy target this recoil energy is 0.5, 5, 20 and 40 keV. This recoil energy is the same for both types of WIMPs, only the rate is different for low WIMP mass.
• The above behavior is carried over to the total rates. For WIMP flows the maximum is in winter, but for the M-B distribution one finds the usual case (maximum in June 3nd) for low reduced mas but maximum in December for relatively large reduced mass. The total rate R 0 for usual WIMP (top panels) and and the scalar WIMP (middle panels) and the relative modulation h (bottom panels) as a function of the WIMP mass in GeV in the case of a heavy target 127 I at zero threshold. The panels on the right are a restriction of those on the left to smaller masses. Otherwise the notation is the same as that of Fig. 5.